Symmetric group on finite set has Sylow subgroup of prime order

From Groupprops
Jump to: navigation, search

Statement

Suppose n is a natural number greater than 1. Let S_n be the Symmetric group (?) of degree n. Then, there exists a prime p such that S_n has a p-Sylow subgroup (?) of order p, i.e., it is a Group of prime order (?).

Facts used

  1. Bertrand's postulate: The version we use states that for any n \ge 2, there exists a prime p such that p \le n < 2p.
  2. Sylow subgroups exist

Proof

The symmetric group of degree n has order n!. By fact (1), there exists a prime p such that n/2 < p \le n. Thus, the largest power of p dividing n! is p. In particular, this means that the order of the p-Sylow subgroup (which exists by fact (2)) is p.